1 Answer
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Let $Y_0 = 3$ and $Y_{i+1} = Y_i^2-2$. Then your $S_i = \frac{1}{2} Y_{2i}$. So your condition would be that $Y_{p-1}\equiv Y_0 \pmod{W_p}$. This is nearly the same as (and I'd say equivalent to) the second conjecture posed here on mersenneforum.org. See the link for some discussion, partial results and variations of the test.